Supercriticality and the Turbulent Energy Cascade in Primitive Equations

A major concern for climate studies is to understand how turbulent eddy fluxes control the atmospheric and oceanic mean state. Held and Larichev (1996) (hereinafter: HL) proposed a theoretical framework to understand the characteristics of the turbulent eddies that arise in an unstable baroclinic flow. Based on the two-layer quasi-geostrophic (QG) model, they derive scaling relations for the spatial scale and energy of the dominant eddies. Both are shown to be controlled by the so-called criticality parameter, which characterizes the supercriticality of the mean flow to baroclinic instability. In particular, they showed that the range of the inverse eddy kinetic energy cascade, from the deformation scale (where EKE is generated) to the Rhines scale (where the cascade is halted), increases proportionally to the criticality parameter.

The goal of this contribution is to examine whether the theory of HL applies to the continuously stratified primitive equations. It has been argued that primitive equation systems, unlike QG models, tend to equilibrate to marginally critical or sub-critical mean states with no significant energy cascade. If this was the case, the theory of HL would be irrelevant for understanding Earth's atmosphere and ocean. In contrast, we will here show that a series of numerical simulations, using a full multi-level primitive equation model, exhibit a large range of supercritical states. The properties of the turbulent flow in the simulations are found to agree well with the theoretical predictions of HL. In particular, we observe an inverse kinetic energy cascade from the deformation scale to the Rhines scale, with the cascade range increasing with the criticality parameter. The theory of HL is then used to obtain a scaling relation for the eddy diffusivity, which can be applied to predict the equilibration of the mean states in the presented simulations.